// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

namespace Eigen {

namespace internal {
template<typename _MatrixType>
struct traits<ColPivHouseholderQR<_MatrixType>> : traits<_MatrixType>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};

} // end namespace internal

/** \ingroup QR_Module
 *
 * \class ColPivHouseholderQR
 *
 * \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
 *
 * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
 *
 * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
 * such that
 * \f[
 *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
 * upper triangular matrix.
 *
 * This decomposition performs column pivoting in order to be rank-revealing and improve
 * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::colPivHouseholderQr()
 */
template<typename _MatrixType>
class ColPivHouseholderQR : public SolverBase<ColPivHouseholderQR<_MatrixType>>
{
  public:
	typedef _MatrixType MatrixType;
	typedef SolverBase<ColPivHouseholderQR> Base;
	friend class SolverBase<ColPivHouseholderQR>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(ColPivHouseholderQR)
	enum
	{
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
	};
	typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
	typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
	typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
	typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
	typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
	typedef HouseholderSequence<MatrixType,
								typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type>
		HouseholderSequenceType;
	typedef typename MatrixType::PlainObject PlainObject;

  private:
	typedef typename PermutationType::StorageIndex PermIndexType;

  public:
	/**
	 * \brief Default Constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
	 */
	ColPivHouseholderQR()
		: m_qr()
		, m_hCoeffs()
		, m_colsPermutation()
		, m_colsTranspositions()
		, m_temp()
		, m_colNormsUpdated()
		, m_colNormsDirect()
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
	}

	/** \brief Default Constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa ColPivHouseholderQR()
	 */
	ColPivHouseholderQR(Index rows, Index cols)
		: m_qr(rows, cols)
		, m_hCoeffs((std::min)(rows, cols))
		, m_colsPermutation(PermIndexType(cols))
		, m_colsTranspositions(cols)
		, m_temp(cols)
		, m_colNormsUpdated(cols)
		, m_colNormsDirect(cols)
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This constructor computes the QR factorization of the matrix \a matrix by calling
	 * the method compute(). It is a short cut for:
	 *
	 * \code
	 * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
	 * qr.compute(matrix);
	 * \endcode
	 *
	 * \sa compute()
	 */
	template<typename InputType>
	explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix)
		: m_qr(matrix.rows(), matrix.cols())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_colsPermutation(PermIndexType(matrix.cols()))
		, m_colsTranspositions(matrix.cols())
		, m_temp(matrix.cols())
		, m_colNormsUpdated(matrix.cols())
		, m_colNormsDirect(matrix.cols())
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
		compute(matrix.derived());
	}

	/** \brief Constructs a QR factorization from a given matrix
	 *
	 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c
	 * MatrixType is a Eigen::Ref.
	 *
	 * \sa ColPivHouseholderQR(const EigenBase&)
	 */
	template<typename InputType>
	explicit ColPivHouseholderQR(EigenBase<InputType>& matrix)
		: m_qr(matrix.derived())
		, m_hCoeffs((std::min)(matrix.rows(), matrix.cols()))
		, m_colsPermutation(PermIndexType(matrix.cols()))
		, m_colsTranspositions(matrix.cols())
		, m_temp(matrix.cols())
		, m_colNormsUpdated(matrix.cols())
		, m_colNormsDirect(matrix.cols())
		, m_isInitialized(false)
		, m_usePrescribedThreshold(false)
	{
		computeInPlace();
	}

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
	 * *this is the QR decomposition, if any exists.
	 *
	 * \param b the right-hand-side of the equation to solve.
	 *
	 * \returns a solution.
	 *
	 * \note_about_checking_solutions
	 *
	 * \note_about_arbitrary_choice_of_solution
	 *
	 * Example: \include ColPivHouseholderQR_solve.cpp
	 * Output: \verbinclude ColPivHouseholderQR_solve.out
	 */
	template<typename Rhs>
	inline const Solve<ColPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	HouseholderSequenceType householderQ() const;
	HouseholderSequenceType matrixQ() const { return householderQ(); }

	/** \returns a reference to the matrix where the Householder QR decomposition is stored
	 */
	const MatrixType& matrixQR() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return m_qr;
	}

	/** \returns a reference to the matrix where the result Householder QR is stored
	 * \warning The strict lower part of this matrix contains internal values.
	 * Only the upper triangular part should be referenced. To get it, use
	 * \code matrixR().template triangularView<Upper>() \endcode
	 * For rank-deficient matrices, use
	 * \code
	 * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
	 * \endcode
	 */
	const MatrixType& matrixR() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return m_qr;
	}

	template<typename InputType>
	ColPivHouseholderQR& compute(const EigenBase<InputType>& matrix);

	/** \returns a const reference to the column permutation matrix */
	const PermutationType& colsPermutation() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return m_colsPermutation;
	}

	/** \returns the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \warning a determinant can be very big or small, so for matrices
	 * of large enough dimension, there is a risk of overflow/underflow.
	 * One way to work around that is to use logAbsDeterminant() instead.
	 *
	 * \sa logAbsDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar absDeterminant() const;

	/** \returns the natural log of the absolute value of the determinant of the matrix of which
	 * *this is the QR decomposition. It has only linear complexity
	 * (that is, O(n) where n is the dimension of the square matrix)
	 * as the QR decomposition has already been computed.
	 *
	 * \note This is only for square matrices.
	 *
	 * \note This method is useful to work around the risk of overflow/underflow that's inherent
	 * to determinant computation.
	 *
	 * \sa absDeterminant(), MatrixBase::determinant()
	 */
	typename MatrixType::RealScalar logAbsDeterminant() const;

	/** \returns the rank of the matrix of which *this is the QR decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index rank() const
	{
		using std::abs;
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
		Index result = 0;
		for (Index i = 0; i < m_nonzero_pivots; ++i)
			result += (abs(m_qr.coeff(i, i)) > premultiplied_threshold);
		return result;
	}

	/** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline Index dimensionOfKernel() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return cols() - rank();
	}

	/** \returns true if the matrix of which *this is the QR decomposition represents an injective
	 *          linear map, i.e. has trivial kernel; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInjective() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return rank() == cols();
	}

	/** \returns true if the matrix of which *this is the QR decomposition represents a surjective
	 *          linear map; false otherwise.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isSurjective() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return rank() == rows();
	}

	/** \returns true if the matrix of which *this is the QR decomposition is invertible.
	 *
	 * \note This method has to determine which pivots should be considered nonzero.
	 *       For that, it uses the threshold value that you can control by calling
	 *       setThreshold(const RealScalar&).
	 */
	inline bool isInvertible() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return isInjective() && isSurjective();
	}

	/** \returns the inverse of the matrix of which *this is the QR decomposition.
	 *
	 * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
	 *       Use isInvertible() to first determine whether this matrix is invertible.
	 */
	inline const Inverse<ColPivHouseholderQR> inverse() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return Inverse<ColPivHouseholderQR>(*this);
	}

	inline Index rows() const { return m_qr.rows(); }
	inline Index cols() const { return m_qr.cols(); }

	/** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
	 *
	 * For advanced uses only.
	 */
	const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

	/** Allows to prescribe a threshold to be used by certain methods, such as rank(),
	 * who need to determine when pivots are to be considered nonzero. This is not used for the
	 * QR decomposition itself.
	 *
	 * When it needs to get the threshold value, Eigen calls threshold(). By default, this
	 * uses a formula to automatically determine a reasonable threshold.
	 * Once you have called the present method setThreshold(const RealScalar&),
	 * your value is used instead.
	 *
	 * \param threshold The new value to use as the threshold.
	 *
	 * A pivot will be considered nonzero if its absolute value is strictly greater than
	 *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
	 * where maxpivot is the biggest pivot.
	 *
	 * If you want to come back to the default behavior, call setThreshold(Default_t)
	 */
	ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
	{
		m_usePrescribedThreshold = true;
		m_prescribedThreshold = threshold;
		return *this;
	}

	/** Allows to come back to the default behavior, letting Eigen use its default formula for
	 * determining the threshold.
	 *
	 * You should pass the special object Eigen::Default as parameter here.
	 * \code qr.setThreshold(Eigen::Default); \endcode
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	ColPivHouseholderQR& setThreshold(Default_t)
	{
		m_usePrescribedThreshold = false;
		return *this;
	}

	/** Returns the threshold that will be used by certain methods such as rank().
	 *
	 * See the documentation of setThreshold(const RealScalar&).
	 */
	RealScalar threshold() const
	{
		eigen_assert(m_isInitialized || m_usePrescribedThreshold);
		return m_usePrescribedThreshold
				   ? m_prescribedThreshold
				   // this formula comes from experimenting (see "LU precision tuning" thread on the list)
				   // and turns out to be identical to Higham's formula used already in LDLt.
				   : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
	}

	/** \returns the number of nonzero pivots in the QR decomposition.
	 * Here nonzero is meant in the exact sense, not in a fuzzy sense.
	 * So that notion isn't really intrinsically interesting, but it is
	 * still useful when implementing algorithms.
	 *
	 * \sa rank()
	 */
	inline Index nonzeroPivots() const
	{
		eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
		return m_nonzero_pivots;
	}

	/** \returns the absolute value of the biggest pivot, i.e. the biggest
	 *          diagonal coefficient of R.
	 */
	RealScalar maxPivot() const { return m_maxpivot; }

	/** \brief Reports whether the QR factorization was successful.
	 *
	 * \note This function always returns \c Success. It is provided for compatibility
	 * with other factorization routines.
	 * \returns \c Success
	 */
	ComputationInfo info() const
	{
		eigen_assert(m_isInitialized && "Decomposition is not initialized.");
		return Success;
	}

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	friend class CompleteOrthogonalDecomposition<MatrixType>;

	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	void computeInPlace();

	MatrixType m_qr;
	HCoeffsType m_hCoeffs;
	PermutationType m_colsPermutation;
	IntRowVectorType m_colsTranspositions;
	RowVectorType m_temp;
	RealRowVectorType m_colNormsUpdated;
	RealRowVectorType m_colNormsDirect;
	bool m_isInitialized, m_usePrescribedThreshold;
	RealScalar m_prescribedThreshold, m_maxpivot;
	Index m_nonzero_pivots;
	Index m_det_pq;
};

template<typename MatrixType>
typename MatrixType::RealScalar
ColPivHouseholderQR<MatrixType>::absDeterminant() const
{
	using std::abs;
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return abs(m_qr.diagonal().prod());
}

template<typename MatrixType>
typename MatrixType::RealScalar
ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
	return m_qr.diagonal().cwiseAbs().array().log().sum();
}

/** Performs the QR factorization of the given matrix \a matrix. The result of
 * the factorization is stored into \c *this, and a reference to \c *this
 * is returned.
 *
 * \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
 */
template<typename MatrixType>
template<typename InputType>
ColPivHouseholderQR<MatrixType>&
ColPivHouseholderQR<MatrixType>::compute(const EigenBase<InputType>& matrix)
{
	m_qr = matrix.derived();
	computeInPlace();
	return *this;
}

template<typename MatrixType>
void
ColPivHouseholderQR<MatrixType>::computeInPlace()
{
	check_template_parameters();

	// the column permutation is stored as int indices, so just to be sure:
	eigen_assert(m_qr.cols() <= NumTraits<int>::highest());

	using std::abs;

	Index rows = m_qr.rows();
	Index cols = m_qr.cols();
	Index size = m_qr.diagonalSize();

	m_hCoeffs.resize(size);

	m_temp.resize(cols);

	m_colsTranspositions.resize(m_qr.cols());
	Index number_of_transpositions = 0;

	m_colNormsUpdated.resize(cols);
	m_colNormsDirect.resize(cols);
	for (Index k = 0; k < cols; ++k) {
		// colNormsDirect(k) caches the most recent directly computed norm of
		// column k.
		m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
		m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
	}

	RealScalar threshold_helper =
		numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
	RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());

	m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
	m_maxpivot = RealScalar(0);

	for (Index k = 0; k < size; ++k) {
		// first, we look up in our table m_colNormsUpdated which column has the biggest norm
		Index biggest_col_index;
		RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols - k).maxCoeff(&biggest_col_index));
		biggest_col_index += k;

		// Track the number of meaningful pivots but do not stop the decomposition to make
		// sure that the initial matrix is properly reproduced. See bug 941.
		if (m_nonzero_pivots == size && biggest_col_sq_norm < threshold_helper * RealScalar(rows - k))
			m_nonzero_pivots = k;

		// apply the transposition to the columns
		m_colsTranspositions.coeffRef(k) = biggest_col_index;
		if (k != biggest_col_index) {
			m_qr.col(k).swap(m_qr.col(biggest_col_index));
			std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
			std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
			++number_of_transpositions;
		}

		// generate the householder vector, store it below the diagonal
		RealScalar beta;
		m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);

		// apply the householder transformation to the diagonal coefficient
		m_qr.coeffRef(k, k) = beta;

		// remember the maximum absolute value of diagonal coefficients
		if (abs(beta) > m_maxpivot)
			m_maxpivot = abs(beta);

		// apply the householder transformation
		m_qr.bottomRightCorner(rows - k, cols - k - 1)
			.applyHouseholderOnTheLeft(m_qr.col(k).tail(rows - k - 1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k + 1));

		// update our table of norms of the columns
		for (Index j = k + 1; j < cols; ++j) {
			// The following implements the stable norm downgrade step discussed in
			// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
			// and used in LAPACK routines xGEQPF and xGEQP3.
			// See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
			if (m_colNormsUpdated.coeffRef(j) != RealScalar(0)) {
				RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
				temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
				temp = temp < RealScalar(0) ? RealScalar(0) : temp;
				RealScalar temp2 =
					temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) / m_colNormsDirect.coeffRef(j));
				if (temp2 <= norm_downdate_threshold) {
					// The updated norm has become too inaccurate so re-compute the column
					// norm directly.
					m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
					m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
				} else {
					m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
				}
			}
		}
	}

	m_colsPermutation.setIdentity(PermIndexType(cols));
	for (PermIndexType k = 0; k < size /*m_nonzero_pivots*/; ++k)
		m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));

	m_det_pq = (number_of_transpositions % 2) ? -1 : 1;
	m_isInitialized = true;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType>
template<typename RhsType, typename DstType>
void
ColPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	const Index nonzero_pivots = nonzeroPivots();

	if (nonzero_pivots == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(rhs);

	c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint());

	m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
		.template triangularView<Upper>()
		.solveInPlace(c.topRows(nonzero_pivots));

	for (Index i = 0; i < nonzero_pivots; ++i)
		dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
	for (Index i = nonzero_pivots; i < cols(); ++i)
		dst.row(m_colsPermutation.indices().coeff(i)).setZero();
}

template<typename _MatrixType>
template<bool Conjugate, typename RhsType, typename DstType>
void
ColPivHouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	const Index nonzero_pivots = nonzeroPivots();

	if (nonzero_pivots == 0) {
		dst.setZero();
		return;
	}

	typename RhsType::PlainObject c(m_colsPermutation.transpose() * rhs);

	m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
		.template triangularView<Upper>()
		.transpose()
		.template conjugateIf<Conjugate>()
		.solveInPlace(c.topRows(nonzero_pivots));

	dst.topRows(nonzero_pivots) = c.topRows(nonzero_pivots);
	dst.bottomRows(rows() - nonzero_pivots).setZero();

	dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>());
}
#endif

namespace internal {

template<typename DstXprType, typename MatrixType>
struct Assignment<DstXprType,
				  Inverse<ColPivHouseholderQR<MatrixType>>,
				  internal::assign_op<typename DstXprType::Scalar, typename ColPivHouseholderQR<MatrixType>::Scalar>,
				  Dense2Dense>
{
	typedef ColPivHouseholderQR<MatrixType> QrType;
	typedef Inverse<QrType> SrcXprType;
	static void run(DstXprType& dst,
					const SrcXprType& src,
					const internal::assign_op<typename DstXprType::Scalar, typename QrType::Scalar>&)
	{
		dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
	}
};

} // end namespace internal

/** \returns the matrix Q as a sequence of householder transformations.
 * You can extract the meaningful part only by using:
 * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
template<typename MatrixType>
typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType
ColPivHouseholderQR<MatrixType>::householderQ() const
{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}

/** \return the column-pivoting Householder QR decomposition of \c *this.
 *
 * \sa class ColPivHouseholderQR
 */
template<typename Derived>
const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::colPivHouseholderQr() const
{
	return ColPivHouseholderQR<PlainObject>(eval());
}

} // end namespace Eigen

#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
